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A fractional calculus of variations for multiple integrals with application to vibrating string. (English) Zbl 1309.49003

Summary: We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler-Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string.{
©2010 American Institute of Physics}

MSC:

49J10 Existence theories for free problems in two or more independent variables
49K10 Optimality conditions for free problems in two or more independent variables
26A33 Fractional derivatives and integrals
26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
49S05 Variational principles of physics
70H03 Lagrange’s equations
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
74K05 Strings
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